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A discrete time system is described by the following system of equations.

$$q[n] = \big(x[n]-\frac k4q[n-1]\big)$$ $$y[n] = \big(q[n]-\frac k3q[n-1]\big)$$

Find the systen function and then find the values of $k$ for which the system is stable. Also find the values of $k$ for which the system is of mimimum-phase.

Can anybody give me some guidelines as to how to handle this system of equations in order to find the system function?

This is an unsolved exercise given by my professor.

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  • $\begingroup$ Read about BIBO stability $\endgroup$ – Maxtron Sep 21 '18 at 15:50
  • $\begingroup$ @Maxtron I don't understand how to use this information to find the transfer function. $\endgroup$ – thelaw Sep 21 '18 at 16:04
  • $\begingroup$ Find the Z-transform. Then find the condition such that the region of convergence is inside the unit circle. $\endgroup$ – Maxtron Sep 21 '18 at 16:13
  • $\begingroup$ this is an interesting convention for the state-variable model: $$\begin{align} \mathbf{q}[n] &= \mathbf{A \, q}[n-1] \ + \ \mathbf{B \, x}[n] \\ \mathbf{y}[n] &= \mathbf{C \, q}[n] \qquad + \ \mathbf{D \, x}[n] \\ \end{align}$$ i sorta like it because it leaves "$x[n]$" as the symbol for the input rather than using that symbol for the state of the system. and it doesn't use "$u[n]$" for the input (which can leave that for the unit step function.) $\endgroup$ – robert bristow-johnson Sep 21 '18 at 21:55
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HINT:

(because it's a homework problem)

  1. Apply the $\mathcal{Z}$-transform to both equations. Use the first equation to express $Q(z)$ in terms of $X(z)$, and plug that into the second equation to obtain $Y(z)$ in terms of $X(z)$. From this you can easily get the transfer function $H(z)$.
  2. Compute the poles and zeros of $H(z)$ and determine the constant $k$ such that the system is stable and/or minimum-phase. You should know what stability and minimum-phase mean in terms of poles and zeros (if not, it should be easy to find that online or in a textbook).
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